Reverse mathematics and order theoretic fixed point theorems

Abstract

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within $\mathsf {RCA_0}$ give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over $\mathsf {RCA_0}$. Then we within $\mathsf {RCA_0}$ give proofs of Knaster–Tarski fixed point theorem, Tarski–Kantorovitch fixed point theorem, Bourbaki–Witt fixed point theorem, and Abian–Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian–Brown least fixed point theorem, Davis’ converse for countable lattices, Markowski’s converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over $\mathsf {RCA_0}$. Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.